Cohomology of p-adic fields and Local class field theory

In this expository article, we outline a basic theory of group (co) homology and prove a cohomological formulation of the Local Reciprocity Law: Gal (L/K) ^ ab HT^-2 (Gal (L/K), Z) HT^0 (Gal (L/K), L^) K^{ Nm₋/₊ (L^) } We first recall basic facts about local fields and homological algebra. Then we define group (co) homology, Tate cohomology, and furnish a toolbox. The Local Reciprocity Law is proven in an abstract cohomological setting, then applied to the case of local fields.